CHAP. I] PERFECT, MULTIPLY PERFECT, AND AMICABLE NUMBERS. 21 the impossibility of which is evident when the exponents /?, 7, . . . are other than 1, 0, 0,. . ., a case giving Euclid's solution [cf. Desboves127]. C. G. Reuschle108 gave in his table C the exponent to which 2 belongs modulo p, for each prime p<5000. Thus 2n-l has the factor 1399 for n = 233, the factor 2687 for rc = 79, and 3391 for n = 113 [as stated explicitly by Le Lasseur119'132]; also 23514513 for ft =47, 1433 for n = 179, and 1913 for n = 239. In the addition (p. 22) to Table A, he gave the prime factors of 2n 1 for various n's to 156, 37 being the least n for which the decomposition is not given completely, while 41 is the least n for which no factor is known. For 34 errata in Table C, see Cunningham110 of Ch. VII. F. Landry109 gave a new proof that 231 1 is a prime. Jean Plana110 gave (p. 130) the factorization into two primes: 241- 1 = 13367X164511353. His statement (p. 141) that 253 1 has no factor < 50033 was corrected by Landry113 (quoted by Lucas,119 p. 280) and Gerardin.177 Giov. Nocco111 showed that an odd perfect number has at least three distinct prime factors. For, if ambn is perfect, 61 a- whence 2(6-1) ~"2(6-l)aw~ 6n+1-l But the minimum values of a, b are 3, 5. Thus b(a 2)>2a 2, a6n-2l[)n = l[)n-1.6(a-2)>6n-1(2a-2), abfl+2bn-1>26n+2a&n-1, contrary to the earlier equation. In attempting to prove that every even perfect number 2wbW ... is of Euclid's type, he stated without proof that hn+l 1 rr+1 1 2m+1bV. . . = (2W+1 -1)5(7. . ., £ = , /, C = ~ - ,. . . o l c 1 require that 2m+1 = J3, bn = 2w+1-l, cf = C, . . . (the first two of which results yield Euclid's formula). F. Landry112 stated (p. 8) that he possessed the complete decomposition of 2n=t=l(n^64) except for 261=tl, 264+l, and gave (pp. 10-11) the factors of 275-l and of 2n+l for n-65, 66, 69, 75, 90, 105. 108Mathematische Abhandlung, enthaltend neue Zahlentheoretische Tabcllen sammt ciner dieselben betreffenden Correspondenz mit dem verewigten C. G. J. Jacobi. Prog., Stuttgart, 1856, 61 pp. Described by Kummer, Jour, fur Math., 53, 1857, 379. l°9Proc6d6s nouveaux pour demontrer que le nombre 2147483647 est premier. Paris, 1850. Reprinted in Sphinx-Oedipe, Nancy, 1909, 6-9. u°Mem. Reale Ac. Sc. Torino, (2), 20, 1863, dated Nov. 20, 1859. luAlcune teorie su'numeri pari, impari, e perfetti, Lecce, 1863. 118 Aux math^maticiens de toutea lea parties du monde: communication sur la decomposition dea nombres en leurs facteure simples, Paris, 1867, 12 pp.